Under the Radar The Ubiquity of Mathematics and
Under the Radar: The Ubiquity of Mathematics and Statistics in University Education Mark Green, UCLA
The Big Questions The role of Math has changed and expanded; what does this mean for the role of Math Depts? What is our vision for Quantitative Education for the 21 st Century? Are we meeting the needs of our students? What should WE, as faculty, do to adapt to this new environment? What information would be helpful in answering these questions?
Quantitative Education for the 21 st Century Do we have a vision as a profession about our educational mission? Such a vision should be nuanced: One size does not fit all The goal is coherence without uniformity An education that provides the foundation for a lifetime— relevant but not trendy How do we build capacity to carry out this vision? What changes do we need to make in our role as faculty to adapt to what such a vision asks of us?
A Few Questions What information do we need that we do not already have? What student populations are we trying to serve? What mathematical content and skills will prepare our students for the world they will live in? Can we shorten the time from discovery to entering the curriculum? What departments should we partner with? How do we move from individual-centered course innovation to systemic change? How to we scale up our successes? Where will the resources come from? How best do we forge a consensus for change?
The “Unreasonable Effectiveness of Mathematics” Prime Numbers-> Secure Internet Commerce Operators on Hilbert Space-> Quantum Mechanics Quaternions->Satellite Tracking, Video Games Eigenvectors-> Google’s Page. Rank Stochastic Processes-> Black-Scholes Integral Geometry-> MRI and PET scans Connections-> Gauge Fields
The “Unreasonable Effectiveness of Applications” Electromagnetics -> Hodge Theory Nuclear Physics -> Random Matrix Theory Geosciences -> Chaotic Dynamical Systems Superconductivity -> Ginzburg-Landau Equation String Theory -> Gromov-Witten Invariants Condensed Matter -> Complex Systems Epidemiology -> Interacting Particle Systems Deep Learning -> ? ? Mathematics evolves. This is one of the ways new “mathematical species” come into existence.
My Subject: The State of Play at UCLA Who requires what? Who teaches what? What are the major trends?
Majors That Require Math + Stat All Life Science majors Computational and Systems Biology Environmental Science Neuroscience Physiological Science Psychobiology Cognitive Science Economics Business Economics Anthropology (BS) Human Biology and Society (BS) Applied Math (but not Math!)
Majors Requiring Math But Not Stat All Physical Science All Engineering Earth and Environmental Science Climate Science Math (tracks except Applied Math)
Majors Requiring Stat But Not Math Psychology Sociology Political Science Public Affairs International Development Studies Communications Human Biology and Society (BA) Anthropology (BA)
Easy to Remember Summary Math + Stat: Life Sciences Math: Physical Sciences and Engineering Stat: Social Sciences Neither: Humanities, Arts, Ethnic and Gender Studies Key Takeaway: We have an enormous responsibility!
Some Trends New majors are proliferating (135 total at UCLA) Majors which are not computational are birthing majors that are: e. g. Psychology Neuroscience, Cognitive Science, Psychobiology; Anthropology and Human Biology and Society have both BA and BS tracks Minors are proliferating (94 total at UCLA), often so students have computational credentials, e. g Digital Humanities minor Lots of Math and Stat courses are being taught in other departments; often these courses marry disciplinary knowledge and quantitative methods Data Science!
Into The Weeds With Mark I want to talk about specific subjects, what courses they offer (not necessarily required) and the trends they illustrate I will try to give an idea of WHY these various subjects need the topics that they do
Ecology and Evolutionary Biology C 119 A: Mathematical and Computational Modeling in Ecology C 119 B: Modeling in Ecological Research 133: Elements of Theoretical and Computational Biology (Intro to core Math ideas and models necessary to…) C 171: Practical Computing for Evolutionary Biologists and Ecologists C 172: Advanced Statistics in Ecology and Evolutionary Biology M 178: Computational Systems Biology: Modeling and Simulation of Biological Systems
Why Population models like dp/dt = r p – a p^2 Lotke-Volterra predator (y)-prey (x) model dx/dt – ax – bxy; dy/dt = cxy-ey; a, b, c, e>0 Modeling effect of ecological diversity on resistance to invasive species (systems of ODE) Modeling extinction, neutral evolution (probabilistic models) Modeling speciation and the role of habitat (spatial statistics) Island Biogeography (power laws, extinction models, spatial statistics, . . ) Statistics of monitoring population distributions
Economics 41: Statistics for Economists 97: Economic Toolkit (“essential math and programming skills, ” e. g. partial derivatives) 106 G: Introduction to Game Theory 106 GL: Introduction to Game Theory Lab 140: Inequality: Mathematical and Econometric Approach 141: Topics in Microecon: Mathematical Finance 142: Topics in Microecon: Probabilistic Microeconomics 143: Advanced Econometrics 144: Economic Forecasting (“theory and applications of timeseries methods) 145: Topics in Microecon: Mathematical Economics 147: Financial Econometrics (reviews probability and statistics)
MS in Business Analytics 400: Mathematics and Statistics for Analytics 403: Optimization 406: Prescriptive Models and Data Analytics 434: Advanced Workshop on Machine Learning 436: Fraud Analytics
MS in Financial Engineering 402: Econometrics 403: Stochastic Calculus 405: Computational Methods in Finance
Why Gathering and modeling economic data, modern portfolio theory (statistics, linear algebra) Growth models (ODE’s) Forecasting (Time-series, machine learning) First welfare theorem, benefits of markets (optimization) Equilibrium models (fixed-point theorems) Modeling markets (stochastic processes, Bachelier, Black-Scholes) Markets with incomplete or asymmetric information (game theory) Income and wealth distribution, firm size distribution (PDE’s, mean field games) Complex investment vehicles (algorithms and computation)
Sociology 111: Social Networks 112: Introduction to Mathematical Sociology 113: Statistical and Computational Methods for Social Research M 118: Simulating Society: Exploring Artificial Communities 191 E: Undergraduate Seminar: Population Growth Models
Sociology Graduate Courses 208 A-B: Social Network Methods 208 C: Machine Learning for Social Scientists 210 A-B-C: Intermediate Statistical Methods 212 A-B: Quantitative Data Analysis 212 C: Study Design and Other Issues in Quantitative Data Analysis M 213 A: Introduction to Demographic Methods M 213 B: Applied Event History Analysis (mostly different models) M 213 C: Population Models and Dynamics 281: Selected Problems in Mathematical Sociology
Political Science 6: Introduction to Data Analysis 6 R: Intro to Data Analysis, Research Version 170 A: Studies in Statistical Analysis of Political Data 171 A: Applied Formal Models: Collective Action and Social Movements (incl game theory, social networks, . . ) Seminar of Politics of Algorithms
Why Analysis of survey data (linear algebra, multivariate statistics) Social networks, science of science (graph theory, network science, probabilistic models, preferential attachment) Emergent social phenomena (probabilistic models, Schelling’s model, collective action model) Prediction (machine learning) Coalitions and competition, emergence of cooperation (game theory, theory of repeated games) Social choice theory (Arrow Impossibility Thm, Gibbard. Satterthwaite Thm)
Chemistry and Biochemistry C 100: Genomics and Computational Biology 110 B: Intro to Statistical Mechanics and Kinetics C 122: Mathematical Methods in Chemistry (linear alg, complex analysis, ODE’s, PDE’s) C 126 A: Computational Methods for Chemistry C 145: Theoretical and Computational Organic Chemistry CM 160 A: Introduction to Bioinformatics CM 160 B: Algorithms in Bioinformatics C 176: Group Theory and Applications in Inorganic Chem M 186: Stochastic Processes in Biochemical Systems
Why Chemical reaction dynamics, metabolic networks (ODE’s) Sequence alignment across species (edit distance, sequence matching algorithms) Genetic sequencing from DNA fragments (Smith-Waterman algorithm, dynamic programming) 3 -dimensional protein structure (Fourier analysis, machine learning) Molecular orbital theory (periodic table, representation theory) Quantum molecular structure (Fourier analysis, numerical analysis) Stochastic behavior of organisms (probabilistic models, stochastic processes) Bioinformatics (Bayesian statistics, Markov chains, hidden Markov models, machine learning)
Mechanical and Aerospace Engineering 82: Mathematics of Engineering 103: Elementary Fluid Mechanics 107: Intro to Modeling and Analysis of Dynamical Systems 150 A: Intermediate Fluid Mechanics (includes Navier. Stokes) 155: Intermediate Dynamics (Lagrangian mechanics, Euler eqn) M 168: Introduction to Finite Element Methods 169 A: Introduction to Mechanical Vibrations 171 A: Introduction to Feedback and Control Systems
Mechanical and Aerospace Engineering (continued) C 175 A: Probability and Stochastic Processes in Dynamical Systems 181 A: Complex Analysis and Integral Transforms 182 B: Mathematics of Engineering 182 C: Numerical Methods for Engineering Applications 184: Introduction to Geometry Modeling (conics, Bezier curves, . . )
Why This is rather typical engineering math A new trend is using machine learning as a short cut to computation
Computer Science 112: Modeling Uncertainty in Information Systems (random variables, Bayes’ Thm, Markov chains, . . ) CM 121: Introduction to Bioinformatics CM 122: Algorithms for Bioinformatics CM 124 L Computational Genetics 145: Introduction to Data Mining M 146: Introduction to Machine Learning 161: Fundamentals of Artificial Intelligence 168: Computational Methods for Medical Imaging
Computer Science (continued) 170: Mathematical Modeling Methods for CS (numerical and symbolic computation, matrix algebra, statistics, optimization, spectral analysis) 180: Introduction to Algorithms and Complexity M 182: Systems Biomodeling and Simulation Basics 183: Introduction to Cryptography M 184, M 185, CM 186 as before
Why Search (Page. Rank algorithm, eigenvectors, stochastic matrices, Perron-Frobenius Theorem) Imaging (PDE’s, heat equation, wavelets, numerical analysis, compressed sensing, machine learning) Cryptography (number theory, lattice theory) Algorithms (graph theory, complexity theory) Dimensionality reduction (linear algebra, singular value decomposition, Johnson-Lindenstrauss Theorem)
UCLA Medical School Now has a Department of Computational Medicine, chaired by a professor from CS
Why Epidemiology (SIR model, ODE’s, spatial statistics, evolutionary models) Individualized medicine (clustering, machine learning) Physiological modeling (ODE’s, PDE’s, agent-based models) Modeling of infections, modeling of drug delivery, cancer modeling (ODE’s, PDE’s) Genetic diseases (phylogenetic methods, parsimony, the , linkage disequilibrium, probabilistic methods) Brain mapping (vector fields, differential geometry, compressed sensing)
UCLA School of Law The Dean, Jennifer Mnookin, trained at MIT Has PULSE (Program on Understanding Law, Science and Evidence) Two professors have a grant to study policy and governance issues in AI (I am part of a group that periodically has lunch to discuss this) Has a class on “Disruptive Technologies” (I was a guest lecturer)
Why Forensic science (many mathematical techniques) Algorithms and the law (explainability, transparency) Regulatory science (agent-based models) Theory of evidence, Jury selection (probability, machine learning)
Digital Humanities Minor “Places project-based learning at the heart of the curriculum” “Students use tools and methodologies such as threedimensional visualization, data mining, network analysis, and digital mapping” “Students have the opportunity to make significant contributions in fields ranging from archaeology and architecture to history and literature. ”
Mathematics Tracks at UCLA Mathematics (11%) Applied Mathematics (32%) Mathematics/Applied Science (1%) Mathematics/Economics (19%) Financial/Actuarial Mathematics (24%) Mathematics for Teaching (1%) Mathematics of Computation (12%) Mathematics: Data Theory (rolling out this Fall)
Mathematics Major Learning Outcomes Strong mathematical content knowledge of single and multivariate differential and integral calculus, and differential equations Ability to synthesize material, solve problems, and think abstractly Familiarity with linear algebra, techniques of proof, and foundations of real analysis Ability to perform basic computer programming, especially in C++
Mathematics: Data Theory Learning Outcomes Understanding of mathematical and statistical bases of most common methods of data science Ability to explain in writing, with examples, how concepts of statistics and mathematics together solve real-world problems involving data Skillfully manage data Development, comparison, and testing of data-driven models to solve problems Understanding and explanation of variability when fitting and interpreting models of real-world systems Carrying out of reproducible data analysis using accepted practices of research community Written and verbal communication of findings of analyses Identification of areas of active research in data science Insightfully address problems concerning ethics of data use and storage, including data privacy and security
Mathematics: Data Theory Learning Outcomes (continued) Demonstrated mastery of concepts and skills of machine learning, modeling and supervised learning, dimension reduction and unsupervised learning, and deep learning Demonstrated familiarity with numerous software tools used in statistical and data science work and research Demonstrated knowledge of mathematical foundations, including pure and applied linear algebra, basic analysis, probability, and optimization theory Study and evaluation of proofs of mathematical and statistical results employed in data theory Work effectively in a team on a data science problem Demonstrated eligibility for graduate study in applied mathematical science or statistical science
What You Should Do Be curious Be proactive Be willing to go outside your comfort zone Get plugged into existing efforts
What We Should Do as a Profession This is the topic for the discussion period ”Nothing” is not the correct answer Time is not our friend
THANKS MAY THE FORCE BE WITH YOU! and
Psychology 100 A: Psychological Statistics 119 S: Neural Basis of Learning and Computing with Neurons (“how neural networks perform”) 142 H: Advanced Statistical Methods in Psych (honors) M 144: Measurement and its Applications 186 B: Cognitive Science Lab: Neural Networks
Computational and Systems Biology M 175: Stochastic Processes in Biochemical Systems M 186: Computational Systems Biology: Modeling and Simulation in Biological Systems
Bioengineering C 175: Machine Learning and Data-Driven Modeling (probability, hidden Markov models, dimensionality reduction, clustering) M 182: Systems Biomodeling and Simulation Basics (mostly first order ODE’s) M 184: Intro to Computational and Systems Biology CM 186: Computational Systems Biology: Modeling and Simulation of Biological Systems
Microbio. , Immunol. and Mol. Gen. Molecular, Cell and Dev. Biol. 103 BL Advanced Research Analysis in Virology (use bioinformatics and Math modeling software on datasets) 104 BL, 109 BL, 150 BL, 187 BL: Advanced Research Analysis in Virology/Developmental Biology/Plant. Microbe Ecology/Genomic Biology 110 L: Integrative Approach to Discovery in Molecular, Cell and developmental Biology (included rigorous quantification and bioinformatics techniques)
Neuroscience M 135: Dynamical Systems Modeling of Physiological Processes C 172: Neuroimaging and Brain Mapping
Physics 105 A-B: Analytic Mechanics 131 -132: Mathematical Methods of Physics (matrices, operators, Fourier series and integrals, complex variable, Riemann surfaces) 160: Numerical Analysis Techniques and Particle Simulations 180 N: Computational Physics and Astronomy Lab
Atmospheric and Oceanic Sc iences M 120: Introduction to Fluid Mechanics 180: Numerical Methods in Atmospheric Science C 182: Data Analysis in Atmospheric and Oceanic Sciences (principal component analysis, time-series, clustering, model validation)
Earth, Planetary and Space Science 171: Advanced Computing in Geosciences (hypothesis testing, incomplete data, formal modeling, probabilistic testing of models)
Geography 166: Environmental Modeling 169: Introduction to Remote Sensing (introduction to digital image processing) M 171: Introduction to Spatial Statistics 172: Remote Sensing: Digital Image Processing and Analysis
Electrical and Computer Engineering 113: Digital Signal Processing (e. g. Fourier transform) 114: Speech and Image Processing Systems Design 131 A: Probability and Statistics 132 A: Introduction to Communications Systems (includes basic probability, hypothesis testing, . . ) 133 A: Applied Numerical Computing 133 B: Simulation, Optimization and Data Analysis 134: Graph theory in Engineering 141: Principles of Feedback Control (e. g. ODE’s) 142: Linear Systems: State Space Approach C 143 A: Neural Signal Processing and Machine Learning M 146: Introduction to Machine Learning
Linguistics 180: Mathematical Structures in Language 185 A-B: Computational Linguistics
Public Affairs 60: Using Data to Learn About Society 70: Information, Evidence and Persuasion 115: Using Quantitative Methods to Understand Social problems and their Potential Solutions
Society and Genetics 105 A: Ways of Knowing in Life and Human Sciences (DNA sequencing, bioinformatics, statistics, . . )
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